associated bundle construction
Let $G$ be a topological group^{}, $\pi :P\to X$ a (right) principal $G$bundle, $F$ a topological space^{} and $\rho :G\to \text{Aut}(F)$ a representation of $G$ as homeomorphisms of $F$. Then the fiber bundle^{} associated to $P$ by $\rho $, is a fiber bundle ${\pi}_{\rho}:P{\times}_{\rho}F\to X$ with fiber $F$ and group $G$ that is defined as follows:

•
The total space is defined as
$$P{\times}_{\rho}F:=P\times F/G$$ where the (left) action of $G$ on $P\times F$ is defined by
$$g\cdot (p,f):=(p{g}^{1},\rho (g)(f)),\forall g\in G,p\in P,F\in F.$$ 
•
The projection ${\pi}_{\rho}$ is defined by
$${\pi}_{\rho}[p,f]:=\pi (p),$$ where $[p,f]$ denotes the $G$–orbit of $(p,f)\in P\times F$.
Theorem 1.
The above is well defined and defines a $G$–bundle over $X$ with fiber $F$. Furthermore $P{\mathrm{\times}}_{\rho}F$ has the same transition functions^{} as $P$.
Sketch of proof.
To see that ${\pi}_{\rho}$ is well defined just notice that for $p\in P$ and $g\in G$, $\pi (pg)=\pi (p)$. To see that the fiber is $F$ notice that since the principal action is simply transitive^{}, given $p\in P$ any orbit of the $G$–action on $P\times F$ contains a unique representative of the form $(p,f)$ for some $f\in F$. It is clear that an open cover that trivializes $P$ trivializes $P{\times}_{\rho}F$ as well. To see that $P{\times}_{\rho}F$ has the same transition functions as $P$ notice that transition functions of $P$ act on the left and thus commute with the principal $G$–action on $P$. ∎
Notice that if $G$ is a Lie group^{}, $P$ a smooth principal bundle^{} and $F$ is a smooth manifold and $\rho $ maps inside the diffeomorphism group of $F$, the above construction produces a smooth bundle. Also quite often $F$ has extra structure^{} and $\rho $ maps into the homeomorphisms of $F$ that preserve that structure. In that case the above construction produces a “bundle of such structures.” For example when $F$ is a vector space and $\rho (G)\subset \mathrm{GL}(F)$, i.e. $\rho $ is a linear representation of $G$ we get a vector bundle^{}; if $\rho (G)\subset \mathrm{SL}(F)$ we get an oriented vector bundle, etc.
Title  associated bundle construction 

Canonical name  AssociatedBundleConstruction 
Date of creation  20130322 13:26:46 
Last modified on  20130322 13:26:46 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  9 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 55R10 
Defines  associated bundle 